Matrix Exponential for defective eigenvalues

Question

I am trying to compute the matrix exponential ($e^{At}$) of the following matrix:

$$\begin{bmatrix}1 &0&0&0\\1&0&0&0\\1&0&0&0\\1&0&0&0\\\end{bmatrix}$$

The eigenvalues were 0 and 1, but they were defective so I was having difficulty finding the diagonal matrix.

I also tried to check if this was a nilpotent matrix, but it was not. I do know that

$$A^n=A \\n=1,2,….$$

I could use this to expand $e^{At}=I+At+{A^2t^2\over 2!}+…$ but I am not sure where to go from there.

Answer 1

Since all positive powers of A are the same as $A$

$$e^{At}=I+At+{A^2t^2\over 2!}+...= I+ (e^t-1)A$$